Change point detection with respect to variance

Sammanfattning: This thesis examines a simple method for detecting a change with respect to the variance in a sequence of independent normally distributed observations with a constant mean. The method filters out observations with extreme values and divides the sequence into equally large subsequences. For each subsequence, the count of extreme values is translated to a binomial random variable which is tested towards the expected number of extremes. The expected number of extremes comes from prior knowledge of the sequence and a specified probability of how common an extreme value should be. Then specifying the significance level of the goodness-of-fit test yields the number of extreme observations needed to detect a change. The approach is extended to a sequence of independent multivariate normally distributed observations by transforming the sequence to a univariate sequence with the help of the Mahalanobis distance. Thereafter it is possible to apply the same approach as when working with a univariate sequence. Given that a change has occurred, the distribution of the Mahalanobis distance of a multivariate normally distributed random vector with zero mean is shown to approximately follow a gamma distribution. The parameters for the approximated gamma distribution depend only on Σ1−1/2 Σ2Σ1−1/2 with Σ1 and Σ2 being the covariance matrices before and after the change has occurred. In addition to the proposed approach, other statistics such as the largest eigenvalue, the Kullback-Leibler divergence, and the Bhattacharyya distance are considered.